Tritangents to Spherical Curves
A Dissertation presented
by
Yury Arkady Sobolev
to
The Graduate School
in Partial Fulfillment of the
Requirements
for the Degree of
Doctor of Philosophy
in
Mathematics
Stony Brook University
August 2015
Stony Brook University
The Graduate School
Yury Arkady Sobolev
We, the dissertation committee for the above candidate for the
Doctor of Philosophy degree, hereby recommend
acceptance of this dissertation
Oleg Viro - Dissertation Advisor
Professor, Department of Mathematics
Anthony Phillips - Chairperson of Defense
Professor, Department of Mathematics
Dennis Sullivan
Professor, Department of Mathematics
Martin Rocek
Professor, Department of Physics
This dissertation is accepted by the Graduate School
Charles Taber
Dean of the Graduate School
ii
Abstract of the Dissertation
Tritangents to Spherical Curves
by
Yury Arkady Sobolev
Doctor of Philosophy
in
Mathematics
Stony Brook University
2015
A generic smooth immersed closed curve in the 2-sphere has a finite number of circles tangent to it in three points. We present a classification of these
tritangent circles and study how each class changes during deformations of
the curve. We relate these classes to finite type invariants of the curve and
derive a relation among the numbers of tritangent circles in the classes and
finite type invariants of the curve.
iii
Contents
List of Figures
v
Acknowledgements
vi
1 Introduction
1
2 Definitions
2.1 Tritangent circles . . . . . . . . . . . .
2.2 Sign . . . . . . . . . . . . . . . . . . .
2.3 Splitting the set of tritangent circles . .
2.4 Spherical geometry . . . . . . . . . . .
2.5 Surface of tangent circles . . . . . . . .
2.5.1 Big front and its singularities .
2.5.2 Local model for big front . . . .
2.5.3 Structure of fronts . . . . . . .
2.5.4 Interpretation of tritangent sign
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3 Moves
3.1 τ -preserving moves . . . . . . . . . . . . . . . . . .
3.1.1 S1,1,1,1 : Quadruple sheet intersection . . . .
3.1.2 S2,1,1 : Cusp edge and two sheet intersection
3.1.3 S2,2 : Two cusp edge intersection . . . . . . .
3.1.4 S3,1 : Swallowtail and sheet intersection . . .
3.1.5 S4 : Swallowtail pair creation/annihilation .
3.2 Self-tangencies . . . . . . . . . . . . . . . . . . . . .
3.2.1 Index of a curve . . . . . . . . . . . . . . . .
3.2.2 Counting circles in a self-tangency . . . . . .
3.2.3 Direct self-tangency . . . . . . . . . . . . . .
3.2.4 Indirect self-tangency . . . . . . . . . . . . .
3.3 Triple points . . . . . . . . . . . . . . . . . . . . . .
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13
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4 Formulas
29
4.1 Extending the index . . . . . . . . . . . . . . . . . . . . . . . 29
4.2 Normalization . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
References
33
iv
List of Figures
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
Swallowtail singularity . . . . . . . . . . . .
Fronts of a generic curve . . . . . . . . . . .
Ruling of the big front . . . . . . . . . . . .
Big front during moves . . . . . . . . . . . .
S1,1,1,1 move . . . . . . . . . . . . . . . . . .
S2,1,1 move . . . . . . . . . . . . . . . . . . .
S2,2 move . . . . . . . . . . . . . . . . . . .
S3,1 move . . . . . . . . . . . . . . . . . . .
S4 move . . . . . . . . . . . . . . . . . . . .
Fronts near a self-tangency . . . . . . . . . .
Circles in a self-tangency move . . . . . . . .
Circles through a point with a given vector .
Circles in a self-tangency perturbation . . .
Extra circles in a self-tangency perturbation
Indirect self-tangencies . . . . . . . . . . . .
Triple points . . . . . . . . . . . . . . . . . .
Indices in a direct self-tangency . . . . . . .
Viviani’s curve . . . . . . . . . . . . . . . .
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7
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32
Acknowledgements
I am eternally grateful to my wonderful advisor, Oleg Viro. Without his
guidance and sage wisdom, I would not have been able to start this thesis.
Without his patience and encouragement, I would not have been able to
finish it. Thank you for everything!
I am also incredibly grateful to my father, Arkady Sobolev, for teaching
me mathematics at a very young age. This put me on a path to where I am
today. I developed a passion for mathematics only because of his instruction,
love, and support.
I would like to thank Chandrika, Nissim, and Jimmy for listening to
early versions of this thesis, for their helpful feedback, strong friendship, and
willingness to put up with all of my idiosyncrasies. I would not have been
able to finish this without you. I would also like to thank Ben, Ilya, JayJay,
Mark, Matt, Nick, and Silvia for their friendship, advice, and always being
there when I needed them.
vi
1
Introduction
Fabricius-Bjerre [6] obtained a formula for the difference between the number
of external double tangents to a planar curve, where the tangencies occur on
the same side of the tangent line, and internal double tangents, where the
tangencies occur on both sides of the line. Ferrand [5] improved upon this by
using finite order invariants introduced by Arnold [3] to split the formula. In
addition to the side of contact, Ferrand’s formulas involved the orientation of
the line and the curve at the point of tangency. Ferrand also derived similar
results for great circles tangent in two points to spherical curves. A similar
problem is to count the number of circles, not necessarily great, that are
tangent to a spherical curve in three points.
For a generic immersed oriented curve in the sphere, there is a finite
number of circles tangent to it in three points. Tangencies are counted with
multiplicity, so osculating circles tangent in an additional point should also
be counted. Each circle is oriented by the cyclic order its tangencies make
with the curve. This splits the set of all circles tangent in three points into
four classes. Let τ i be the signed count of the circles tangent in three points
plus half the number of circles osculating in one point and tangent at another,
where the orientation at exactly i tangencies agrees with that of the curve.
These numbers turn out to be invariants of the curve of order two.
The image of a spherical curve is a graph on the sphere. The double
points are the vertices, the arcs between the double points are the edges, and
the connected components of the complement are the faces. Each face has
an index, which measures how many times the curve winds around it. The
index may be extended to a function on the edges and vertices by averaging
over adjacent faces. Let F , E, and V be the sums of the index cubed over
each face, edge, and vertex, respectively. An appropriate linear combination
of these numbers recovers the τ i .
2
1
τ 0 = −τ 3 = − F + E − V
3
3
1
2
τ 1 = −τ 2 = F − E + V
3
3
In Section 2, we classify circles tangent to a spherical curves in three
points. We define a sign which determines each circle’s contribution to its
class. Finally, we introduce a surface with singularities which is the moduli
1
space of circles tangent to a curve. In Section 3, we enumerate the deformations of a curve which affect tritangent circles. We also define the index of a
point in the sphere with respect to a curve. Finally, in Section 4, we derive
the combinatorial formulas presented above.
2
2
2.1
Definitions
Tritangent circles
Let γ : S 1 → S 2 be an immersed oriented spherical curve. The tangency
between a circle and a curve is characterized by the order of contact, k. If
the order is 1, the tangency is simple. If k > 1, the tangency is osculating.
If γ is generic, there is a finite number of circles that are simply tangent
to it in exactly three points. We will call such a circle a simple tritangent.
Tangencies should be counted with multiplicity. A simple tangency counts
as a single tangency. However, a second order tangency should be counted
as two tangencies. The curve γ also has a finite number of circles tangent
in two points where one of the tangencies is second order and the other one
is simple. We call such a circle an osculating tritangent. Finally, at each
vertex (extremum of curvature) of a curve, the osculating circle makes third
order contact with the curve. This type of tangency has multiplicity three
and we call the circle a vertex tritangent.
A simple tritangent circle has a natural orientation such that starting at
any tangency point, the next mutual tangency point on the circle following
its orientation is the same as the next mutual tangency point on γ following
the orientation of γ. An osculating or vertex tritangent circle inherits its
orientation from the curve at the point of osculation.
2.2
Sign
An oriented circle C separates the sphere into two sides. The orientation of
the circle and the orientation of the sphere allow us to call one of the sides
left and the other right. Let t be a velocity vector for C at x. Then, a vector
v at x points to the left if the frame (t, v) gives the orientation of the sphere.
At each simple tangency point in a tritangent, the curve is locally on one
side of the tritangent. If the curve is on the left, we call the tangency left.
Otherwise, the tangency is right.
We introduce a sign to distinguish different tritangents. The sign of a
simple tangency is positive if the tangency is left and negative if it is right.
The sign of a simple tritangent is the product of the signs of its three
tangencies. The sign of an osculating tritangent is the opposite of the
sign of the non-osculating simple tangency. At an osculating tangency, the
curve crosses from one side of the osculating cirlce to the other, so we can
3
view it as the limiting behavior of two tangencies, one form the left and one
from the right, coming together and merging. We will use σ(C) to denote
the sign of a tritangent.
2.3
Splitting the set of tritangent circles
Given an oriented circle C and a curve γ tangent to it, the orientations of C
and γ may agree or disagree at the point of tangency. If they agree, we say
the tangency is direct. Otherwise, it is indirect.
The set of simple tritangent circles splits into four subsets according to
how many tangencies are direct. Let T i be the set of simple tritangent circles
with exactly i direct tangencies. We define the following quantities
ti =
σ(C)
C∈T i
The set of osculating tangencies splits in a similar manner. Let S − , S +
be sets of osculating tritangents with the non-osculating tangency indirect
and direct, respectively. Again, we define the signed sum
s± =
σ(C)
C∈S ±
For the purposes of counting tritangents, we need to consider a linear
combination of the numbers of simple and osculating tritangents. We define
τ 0 = t0
τ 1 = t1
s−
τ =t +
2
+
s
τ 3 = t3 +
2
2
2
An osculating tangency may be considered as the merging of two tangencies, one from the left and one from the right, both of which are direct, so
they only contribute to τ 2 and τ 3 . The reason for the factor of 12 will become
clear when we examine deformations in Section 3.
4
2.4
Spherical geometry
The full Möbius group is the group of orientation-preserving and orientationreversing conformal transformations of the sphere. These are Möbius transformations as well as reflections about circles. Under each transformation,
circles are sent to circles. Each transformation is a diffeomorphism, and
therefore preserves order of contact between curves. Furthermore, the side
of contact is preserved. While a reflection with respect to a tangent circle
reverses the side of contact, it also reverses the orientation of the sphere.
The directness of a tangency is clearly preserved. It follows that the τ i are
invariant under the action of this group.
Each oriented circle in the sphere has a signed curvature. Let S 2 be the
unit sphere in R3 and C a circle in the sphere. Such a circle has a radius, r,
as measured in R3 . The unsigned geodesic curvature is defined by
√
1 − r2
kg (C) =
r
This measures how far a circle is from being a great circle. If C is oriented,
we can improve this definition by introducing a signed curvature, κ. The
circle C bounds two disks in S 2 . These disks inherit an orientation from S 2 .
Each disk also induces an orientation on C. Let D be the disk such that
the orientation it induces on the boundary agrees with the given orientation
on C. If C is a great circle, define its signed curvature to be 0. Otherwise,
define
2π − area D
κ(C) =
kg (C)
|2π − area D|
Finally, we define the signed curvature of a curve at a point as the signed
curvature of the osculating circle to the curve at that point.
2.5
Surface of tangent circles
2.5.1
Big front and its singularities
Denote by M , the space of unoriented circles in the sphere. This is the open
unit ball in R3 blown up at the origin. Consider the unit sphere S 2 embedded
in R3 . Any plane that intersects this sphere does so in a circle or a point.
Conversely, any circle in the sphere determines the plane that cuts it out.
Let B be the unit ball in R3 . Then any point x ∈ B \ 0 can be seen as a
5
vector v from the origin and uniquely defines the plane passing through x
and normal to v. Furthermore, every plane not passing through the origin
arises in this way. So, the circles in the sphere which are not great circles are
in a one-to-one correspondence with the points of B \ 0. Great circles are in
a one-to-one correspondence with planes through the origin, and these are
defined by their normal directions. But, we only need the normal direction
without regard for sign since the plane through the origin and normal to a
vector w is the same as the plane normal to the vector −w.
Another description of M is the tautological line bundle over RP 2 . The
zero section is the space of great circles and the fiber over each point is the
set of circles with the same center on the sphere as the center of the base
point great circle.
There is a natural compatification, M , obtained by adding the circles of
radius 0. This is the closed unit ball blown up at the origin with the circles
of radius 0 making up the points of the boundary sphere.
Let γ be a spherical curve parametrized by arclength. Fix a point x on γ.
Let Cx be the family of oriented circles tangent to γ at x. The orientation of
each circle is given by the velocity vector of γ at x. Each circle in this family
is determined by its signed
curvature λ.
The disjoint union x∈γ Cx is a cylinder. We can give it coordinates (s, λ),
where s is the arclength (modulo the total length of the curve) and λ is the
signed curvature. Then, let μ : S 1 × R → M be the map that sends circles
tangent to γ to their corresponding points in M . For a fixed λ0 , the image of
μ(s, λ0 ) is a front of the original curve. The full image S = μ(S 1 ×R) is called
the big front. It follows that μ is a Legendrian map and its singularities are
well classified [2].
Theorem 1. The differential dμ has full rank at C if C is not an osculating
circle. It is rank 1 if C is an osculating circle.
We will prove this theorem below. The map μ is never an immersion as it
fails to have full rank at osculating circles, where it falls by 1. In fact, away
from the vertices of the curve (points where the curvature has an extremum)
the image of the set of osculating circles is a cusp edge. Since every point
on γ has a single osculating circle, the cusp edge is the image of a circle.
However, it is not diffeomorphic to one. At vertices of γ, the image of μ has
a swallowtail.
A neighborhood of a vertex of a curve looks like a parabola. The vertex
of the parabola is the vertex of the curve (hence the name). The centers of
6
Codimension 2
• S1,1,1 : Triple point (simple tritangent)
• S2,1 : Cusp edge / smooth sheet intersection (osculating tritangent)
• S3 : Swallowtail point (vertex tritangent)
Finally, the S1 part of the surface has a coorientation. Let C be a circle
tangent to γ that is not osculating. If we perturb C in a direction transverse
to the curve, it will locally intersect γ in two points or in no points. Perturb
it so that it intersects it in two points. This produces a new circle C that
lies on one side of the big front. We will call this side positive.
2.5.2
Local model for big front
Let p be any point in the complement of γ. We may treat it as the south
pole and stereographically project the rest of the sphere to the plane tangent
at the north pole. Circles tangent to γ in the sphere will map to circles or
lines tangent to the image of γ in the stereographic projection.
The space of circles (but not lines!) in the plane can be identified with
the upper half space M = (R3 )+ . Given a circle C with center (x, y) and
radius r, we send it to the point (x, y, r).
Much like in the sphere, there is a signed planar curvature. It agrees with
the usual notion of curvature, except it is negative if the tangent vector to the
curve is turning clockwise as we move along the curve. The circles tangent
to γ in the stereographic projection are thus parametrized by (s, λ) where s
is the arclength (modulo the total length) and λ ∈ R∗ is the signed planar
curvature. So, the set of circles tangent to γ in the plane is the disjoint union
of two cylinders. Let ν be the map that sends them to M .
Let S be the image of ν inside M . This pair (M , S ) is a local model
for the pair (M, S) away from the projection point p. In fact, we have the
following obvious lemma.
Lemma 1. Let E ⊂ M be the set of circles tangent to p. There exists a
diffeomorphism of pairs, (M \ E, S \ E) → (M , S ).
Proof of Theorem 1. It suffices to prove the theorem locally. Let U be a sufficiently small neighborhood of C in M . We may choose a point p in the sphere
8
such that there are no circles through p that lie in U . We stereographically
project from p and pass to the local model.
Let γ be the stereographic projection of the curve in the sphere. The
space of all circles tangent to γ is parametrized by (s, λ), where s is the
arclength and λ is the signed planar curvature. Denote by κ(s) the signed
planar curvature of γ(s). Let T (s) be the unit tangent vector to γ and N (s)
the unit normal vector. The circle C is mapped to some circle in the plane
tangent to γ. Suppose it is tangent at γ(s0 ) and has planar curvature λ0 .
We restrict dν to the level sets of constant s. Let G be the set of circle
tangent to γ at s0 . The center of such a circle is a function of its curvature
λ and is given by
1
γ(s0 ) + N (s0 )
λ
Differentiating this with respect to λ we get −1
. Since our curvature is never
λ2
0 or ∞, this derivative is never 0 or ∞ either. This implies that the rank of
dν is always at least 1.
On the other hand, if we restrict dν to the level sets of constant λ, the
rank may drop. Let H be the set of circles tangent to γ with curvature λ0 .
The center of each circle in H is given by
1
N (s)
λ0
γ(s) +
We differentiate this with respect to s and apply the Frenet-Serret formula
N = −κT . The derivative is
T (s) −
κ
T (s)
λ0
Since our curve is smooth, T is never 0. It follows that this quantity vanishes
if and only if κ = λ0 . In other words, the rank of the differential drops by 1
only at osculating circles.
2.5.3
Structure of fronts
A horizontal section of the big front at height r is two curves parallel to γ
and offset by a distance r to the right and to the left. These are fronts of γ.
One of these has smaller curvature in absolute value than γ at corresponding
points. It always looks like γ. The other curve has a greater curvature in
absolute value at corresponding points. For small r, it also looks like γ. As
9
Suppose C bounds a disk that has the same orientation as the plane. By
Lemma 2, the tangency at any of the points x, y, or z is right if and only if
the vertical component of the coorienting normal points up. Since the normal
vector is determined by its vertical component, the orientation of the frame
is determined by the sides of tangency.
On the other hand, if C has the opposite orientation, we see that a tangency is right if and only if the vertical component points down. This means
the coorienting normals point in the opposite direction to the case above.
However, since C has the opposite oriention, the order of the vectors in the
frame changes. It follows that the orientation of the frame depends only on
the sides of tangency, and not on the orientation induced on the tritangent
circle.
The frame in (R3 )+ given by (∂x , ∂y , −∂z ) is the positive one. It is easy to
check that a small counter-clockwise oriented circle with three right tangencies corresponds to a negatively oriented frame. The theorem follows.
12
3
Moves
Let Ω be the space of all smooth spherical curves with the Whitney topology.
It has two connected components corresponding to the regular homotopy
classes of curves in the sphere.
A smooth immersed curve may have singularities in the form of selftangencies and triple points. These are singularities of the immersion. A
generic curve has no self-tangencies and no triple points. Generic curves form
an open dense subset in the space of all curves. Curves which violate one
of the above conditions make up a statified hypersurface with singularities
called the discriminant.
The big front for a non-generic curve may have additional singularities.
These are classified in [2]. We will examine each one in the next section.
A one-parameter family of curves is said to be generic if at most a finite
number of its members have one of the degeneracies listed above and no
member has more than one degeneracy. When a family of curves passes
through a degeneracy, the curve undergoes a move. In particular, there are
two perturbations of a degenerate curve. One of these will be called the
negative deformation, and the other the positive deformation. The direction
of the move is the transition from the negative deformation to the positive
one. The directions we choose are based on the ones introduced by Arnold
[3].
When a family of curves passes through a self-tangency or a triple point,
the numbers τ i may change. However, the τ i are invariant under the remaining moves.
3.1
τ -preserving moves
For a generic curve, a circle is said to be tritangent to it if it is tangent in
three points. As we have seen, these three points may be allocated between
simple tangencies, osculating tangencies (which count as two tangencies), and
vertex tangencies at vertices (which count as three). For a one parameter
family, there will be members with circles which are tangent in four points.
In a sense, these form a stratum of codimension −1 in S. The different ways
this can occur is enumerated below. Figure 4 shows the big front at the
moment of each move. In each case, the numbers τ i do not change.
13
the curve γy . This naive definition is not invariant under different choices of
y.
A curve in the plane has an integer regular homotopy class. This is called
the rotation number or turning number of the curve, and we will denote it
by ω. It is defined as the degree of the tangential Gauss map from the curve
to the unit circle which sends each point to the unit velocity vector at the
point as the curve is traversed in the direction of its orientation. We have
the following theorem due to Arnold [1].
Theorem 3. The following quantity is independent of choice of y.
indy (x) −
ω(γy )
2
If we choose y and x to lie in the same connected component of the
complement, then indy (x) = 0. We are motivated to make the following
definition.
ω(γx )
ind(x) = −
2
We will no longer need the planar index, so ind will always refer to the
spherical version. This function is 12 Z valued.
Observe that ind(x) jumps by 1 when x is moved from one connected
component to an adjacent one. Let A and B be adjacent components of the
complement and β a path from a point in A to a point in B that crosses a
mutual boundary component only once. Since the boundary component is
an arc of γ, it is oriented. If it crosses β from left to right, then ind(x) jumps
by +1. If it crosses from right to left, then ind(x) jumps by −1. This tells
us that ind(x) is completely determined by its value on a single point in the
complement of the γ.
3.2.2
Counting circles in a self-tangency
When a curve undergoes a self-tangency move, tritangent circles both appear and disappear. Denote the two branches in the move by L and R and
the point of contact by x. Using a Möbius transformation followed by an
appropriate stereographic projection, we may arrange so the branches have
opposite planar curvature sign and the L branch is on the left.
Theorem 4. Let C be a circle through x and tangent to both branches. Suppose C is simply tangent to the curve in exactly one other point on branch Y .
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If the curvature of C is such that C is locally between L and R in a neighborhood of x, there exist two tritangent circles close to C in the negative
deformation, but not the positive one. Otherwise, if C is locally to one side
of both L and R, there exist two tritangent circles close to C in the positive
deformation, but not the negative one.
Proof. In the planar model for the big front, a self-tangency of the curve
corresponds to two rays of self-tangency of the surface. The two rays emerge
from the self-tangency point at an angle of π4 with the horizontal. A circle C
that passes through x, is tangent to both branches, and is tangent at another
point of the curve at y corresponds to the transverse intersection between
one of these rays and another sheet of the surface. We focus on one of the
rays since the picture is symmetric.
Let FL (h) and FR (h) be fronts of L and R a distance h to the right of
their respective branches. These two fronts are tangent for all h and the
points of tangency form a ray, ρ(h). Let ρ(o) be the osculating circle for R
at x. The neighborhood of ρ is pictured in Figure 3. Below o, FR (h) has the
same curvature sign in a neighborhood of ρ(h) as the branch R. Above o,
the curvature sign is opposite. Furthermore, for h > o, since the center of
curvature of L at x is to the left of the center of curvature of R at x, locally,
the curvature of FL (h) is always less in absolute value than the curvature of
FR (h) (see Figure 10).
As the move resolves either to the negative or positive deformation, FL (h)
intersects FR (h) in two points. This means ρ bifuracates into two pieces of
self-intersection of the surface. So, a third sheet which transversally intersects
ρ will acquire two triple points. If the curvature of C is such that it is locally
to the right of R, the sheet transverse to ρ intersects it below o. Otherwise,
it intersects it above.
The appearing and disappearing circles in the theorem above are illustrated in Figure 11. We need a method of counting these. Let C be a circle
simply tangent to γ at some point. The tangency is either left or right and
either direct or indirect. We use l to denote left tangencies and r to denote
right tangencies. We also use a superscript + to denote direct tangencies
and a superscript − to denote indirect tangencies. For example, an indirect
left tangency is denoted by l− .
For any point x ∈ S 2 \ γ and any tangent vector v at x, we define L−
v
as the set of circles through x, tangent to and oriented by v, and simply
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move effects the following change in the τ i .
Δτ 1 = −Δτ 2 = 4 ind(x) − 4
If the self-tangency is right-handed, the move effects the following change in
the τ i .
Δτ 1 = −Δτ 2 = 4 ind(x) + 4
We follow the same procedure as for a direct self-tangency. When we
choose v, we can choose it to be either vertical direction. The count of
circles is not affected. The following table summarizes the results.
Tangency at y
l−
r−
l+
r+
Δτ 1
1
−1
−1
1
Δτ 2
−1
1
1
−1
We need to account for the extra circles. For a left-handed tangency,
there are two circles passing through x and tangent to the L branch. These
make l− and r+ contact with γ − . There are two circles tangent to the R
branch. These also make l− and r+ contact. So, we are overcounting by 4
circles. For a right-handed tangency, there are two circles passing through x
and tangent to the L branch. These make l+ and r− contact with γ − . There
are also two circles tangent to the R branch. These also make l+ and r−
contact. So, we are undercounting by 4 circles.
3.3
Triple points
Consider a degenerate curve where three branches pass through the same
point. This can be perturbed in two ways. In one deformation, the inscribed
circle between the three branches has an odd number of direct tangencies.
In the other deformation, it has an even number. We choose the one with
the even number to be positive.
This move can happen in four distinct ways. Consider the triangle formed
by the branches of the curve which vanishes during the move. This triangle
has an inscribed circle which is oriented by the order the curve passes the
three branches. The circle bounds a disk contained in the vanishing triangle.
If the orientation induced on this disk from the boundary circle is the same
as the orientation of the sphere, then we say the move is proper. Otherwise,
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Triple point
Proper Strong
Reflected Strong
Proper Weak
Reflected Weak
Δτ 0
−1
1
1
−1
Δτ 1
−3
3
−1
1
Δτ 2
3
−3
1
−1
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Δτ 3
1
−1
−1
1
4
Formulas
We have analyzed how the numbers τ i change under various moves. We will
now derive an explicit formula for them. We do this by exhibiting a function
that has the same jumps during curve moves.
4.1
Extending the index
The double points and arcs of a spherical curve γ form a graph on the sphere.
The complement of the curve is separated into faces. The index was defined
in Section 3.2.1 as a function on these faces. This definition extends to the
graph as follows. Assign to each edge the average of the two adjacent faces
and to each vertex the average of the four adjacent faces. Viro [4] constructs
finite type invariants out of sums of powers of the index over the faces, edges,
and vertices of such a graph. We will use a similar strategy to obtain formulas
for the τ i .
We define the following sums.
F =
ind(f )3
f ∈ faces
E=
ind(e)3
e∈ edges
V =
ind(v)3
v∈ vertices
Theorem 7. For any move,
1
2
Δτ = −Δτ = Δ − F + E − V
3
3
1
2
1
2
Δτ = −Δτ = Δ F − E + V
3
3
0
3
The numbers F , E, and V are invariant under any homotopy of the curve
which does not pass through self-tangencies or triple points. To determine
how these numbers change during a self-tangency or triple point move, we
pick a point in the complement of the curve and fix an index of i for it. As
long as a branch of the curve does not cross over the point during the move,
its index will remain unchanged. So, it suffices to check how the numbers
change locally in a small neighborhood of the move.
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4.2
Normalization
The previous theorem shows that the jumps in the τ i and in the given linear
combinations of F , E, and V agree. This means they differ by at most a
constant. We strengthen the result by showing that the constant is 0.
Theorem 8.
2
1
τ 0 = −τ 3 = − F + E − V
3
3
1
2
τ 1 = −τ 2 = F − E + V
3
3
There are two regular homotopy classes of curves in the sphere. These
are represented by the circle and the figure eight. To prove the theorem, it
suffices to verify it for one example in each class.
Lemma 4. The ellipse has no simple or osculating tritangents.
Proof. Since stereographic projection preserves tangencies, we may work in
the plane. Both the ellipse and circle are cut out by degree two polynomials.
By Bezout’s Theorem, they intersect in at most 4 points, counted with multiplicty. A tangency counts as two points and an osculating tangency counts
as three. A simple tritangent would need to have 6 point tangency and an
osculating tritangent would need to have 5 point tangency.
Lemma 5. There exists a figure eight curve with no simple or osculating
tritangents.
Proof. Consider a unit sphere in R3 resting with its south pole on the origin
of the xy-plane. Viviani’s curve is obtained by intersecting this sphere with
a cylinder of radius 12 , standing on the xy-plane, and passing through the
center of the sphere (see Figure 18). This is a figure eight curve in the
sphere. When projected down along the z direction, it forms a circle C of
radius 12 . Any circle in the sphere will project to an ellipse or a chord of
the unit circle. Suppose there was a tritangent, either simple or osculating.
Projection preserves tangencies, and never decreases the order of contact.
So, the image of the tritangent would be either an ellipse or a chord that
is tangent to C. The image cannot be a chord because the preimage of the
chord is a circle which is simply tangent to the figure eight in at most two
points. The image cannot be an ellipse because it would have to be tangent to
C in at least 5 points, counted with multiplicity. This would violate Bezout’s
Theorem.
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References
[1] V.I. Arnol’d, “The geometry of spherical curves and the algebra of quaternions.” RUSS MATH SURV 50.1 (1995): 1-68.
[2] V.I. Arnol’d, S.M. Gusein-Zade, A.N. Varchenko, “Singularities of differentiable maps”, vol. 1, Birkhuser (1985) (translated from Russian):
285-346.
[3] V.I. Arnol’d, “Plane curves, their invariants, and classifications.” Advances in Soviet Mathematics 21 (1994): 33-91.
[4] O.Y. Viro, “Generic immersions of the circle to surfaces and the complex
topology of real algebraic curves.” Topology of Real Algebraic Varieties
and Related Topics, Amer. Math. Soc. Transl. Ser. 2, vol. 173, American
Mathematical Society (1996): 231-252.
[5] E. Ferrand, “On the Bennequin invariant and the geometry of wave
fronts.” Geom. Dedicata, 65, no. 2 (1997): 219-245.
[6] F Fabricius-Bjerre, “On the double tangents of plane closed curves.”
Math. Scand, 11 (1962): 113-116.
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